In fact, the days of the week in Japanese do refer to the planets,
and the correspondence is the same as in Western languages. In Japanese,
every day name contains the element "you", which means
"luminary, shining body", i.e planet. For example, "getsuyoubi"
is the day (bi) of the planet (you) Moon (getsu).

A bit of history. The Chinese named the planets after the five
elements: water for Mercury, metal for Venus,
fire for Mars, wood for Jupiter, earth
for Saturn. Then someone decided to name the days of the week after the
planets, including the Sun and the Moon, and to use the same same
day-to-planet correspondence as in the West. At the end of the XIXth
century, the Japanese borrowed this convention from the Chinese. The
Chinese themselves abandoned it and use a simple numerical system
today.

In the West, the correspondence between planets and days of the week
was probably invented by the Babylonians, so maybe the Chinese borrowed
it from the Babylonians. An interesting fact is that, while most modern
Western languages have partially lost this correspondence, it has remained intact in
Japanese. The following table lists the name of the days in several
languages. Names surrounded by parenthesis do not correspond to planets: